Axiomatizations of signed discrete Choquet integrals

We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lov\'asz extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory

Decomposition approaches to integration without a measure

Extending the idea of Even and Lehrer [3], we discuss a general approach to integration based on a given decomposition system equipped with a weighting function, and a decomposition of the integrated function. We distinguish two type of decompositions: sub-decomposition based integrals (in economics linked with optimization problems to maximize the possible profit) and super-decomposition based integrals (linked with costs minimization). We provide several examples (both theoretical and realistic) to stress that our approach generalizes that of Even and Lehrer [3] and also covers problems of linear programming and combinatorial optimization. Finally, we introduce some new types of integrals related to optimization tasks.Comment: 15 page

Dealing with Interaction Between Bipolar Multiple Criteria Preferences in PROMETHEE Methods

In this paper, we consider the bipolar approach to Multiple Criteria Decision Analysis (MCDA). In particular we aggregate positive and negative preferences by means of the bipolar PROMETHEE method. To elicit preferences we consider Robust Ordinal Regression (ROR) that has been recently proposed to derive robust conclusions through the use of the concepts of possible and necessary preferences. It permits to take into account the whole set of preference parameters compatible with the preference information provided by the Decision Maker (DM)

Intangibles mismeasurements, synergy, and accounting numbers : a note.

For the last two decades, authors (e.g. Ohlson, 1995; Lev, 2000, 2001) have regularly pointed out the enforcement of limitations by traditional accounting frameworks on financial reporting informativeness. Consistent with this claim, it has been then argued that accounting finds one of its major limits in not allowing for direct recognition of synergy occurring amongst the firm intangible and tangible items (Casta, 1994; Casta & Lesage, 2001). Although the firm synergy phenomenon has been widely documented in the recent accounting literature (see for instance, Hand & Lev, 2004; Lev, 2001) research hitherto has failed to provide a clear approach to assess directly and account for such a henceforth fundamental corporate factor. The objective of this paper is to raise and examine, but not address exhaustively, the specific issues induced by modelling the synergy occurring amongst the firm assets whilst pointing out the limits of traditional accounting valuation tools. Since financial accounting valuation methods are mostly based on the mathematical property of additivity, and consequently may occult the perspective of regarding the firm as an organized set of assets, we propose an alternative valuation approach based on non-additive measures issued from the Choquet's (1953) and Sugeno's (1997) framework. More precisely, we show how this integration technique with respect to a non-additive measure can be used to cope with either positive or negative synergy in a firm value-building process and then discuss its potential future implications for financial reporting.Financial reporting; accounting goodwill; assets synergy; non-additive measures; Choquet’s framework;

Indicators for the characterization of discrete Choquet integrals

Ordered weighted averaging (OWA) operators and their extensions are powerful tools used in numerous decision-making problems. This class of operator belongs to a more general family of aggregation operators, understood as discrete Choquet integrals. Aggregation operators are usually characterized by indicators. In this article four indicators usually associated with the OWA operator are extended to discrete Choquet integrals: namely, the degree of balance, the divergence, the variance indicator and Renyi entropies. All of these indicators are considered from a local and a global perspective. Linearity of indicators for linear combinations of capacities is investigated and, to illustrate the application of results, indicators of the probabilistic ordered weighted averaging -POWA- operator are derived. Finally, an example is provided to show the application to a specific context